Making sense of vector derivatives

Im trying to understand helmholts decomposition, and in order to do so, I feel the need to understand the different ways to apply the del operator to a vector valued function. The dot product and the cross product between two ordinary vectors are easy to understand, thinking about them as a projection onto another vector, or the area of the parallelogram between the vectors as magnitude with the resulting vector perpendicular to both original vectors.
However, when we use the del vector, which is a vector of operators, this way of thinking gets harder. Is there a way of understanding these vector calculus operations by knowing about the geometrical properties of the dot and cross product, which in turn lends better understanding to the second order derivation operations on a vector functions? I know what curl and div is, that's not the problem, Im very interested in an analogy between whats happening with real vectors that's multiplied in these ways vs when the del operator is multiplied in these ways with a vector function. Thanks for any answer!!

They are only written that way as a mnemonic to remember how to calculate them. To understand what they really are you need to understand the coordinate free definitions of them. Divergence of a vector field is is the limit as the volume goes to zero of the flux of a vector field through a closed surface enclosing said volume. Symbolically:

limV→0 1/V∫∂VF*nda.

This is telling you that you can think of the divergence as "the flux at a point". The curl's coordinate free definition is the limit as the area goes to zero of the line integral of a vector field around the curve enclosing said area. Now you should recall that a line integral produces a scalar, but the curl is a vector field, this definition is for the normal component of the curl (normal to the surface in question). Symbolically:

limA→01/A∫∂AF*dr=(∇xF)*n

What this is telling you is that the curl of a vector field "measures" the circulation of the vector field about a point.
So the divergence is sort of "the flux at a point" and the curl is, loosely speaking, "the circulation about a point"
Of course these definitions suppose you know what line and surface integrals are. You should also note that all of these definitions involve spaces and their boundaries. There is a very important relation between the boundary of "something" and its "interior".

By "making sense," I believe you are asking for an intuitive understanding of the vector operations. I asked myself this same question last year, since I try to make sense of all math lore I learn, and I believe this site does a marvelous job of explaining them. Also note, this is not probably where you should start; the site has many other pages progressing towards the definitions, theorems, and explanations. I suggest you look through them, preferably in sequence. The site is easily navigable with the side bar on the right.